The urgency here is a bit overstated - it's not about some emergency call in the middle of the night, but about routine maintenance that's best done in autumn, before the heating season starts. Batteries and pipes really do need attention, but more from a preventive standpoint.
The main issue is that over time, rust, air pockets, and sediment accumulate in heating systems, which reduces heat output. Radiators start heating unevenly or stop working altogether on lower floors. Pipes can rust from the inside, corrosion develops, tiny cracks appear, and eventually this leads to leaks. If your apartment floods your downstairs neighbors - then it all goes through courts and insurance claims, which is expensive and takes forever. Plus, sometimes the system can freeze in an unheated part of the building or in the basement, and then the entire riser stops working.
The check itself is simple: turn on your radiator around September-October and touch it at the bottom and top - it should be hot everywhere. If the top is cold, there's air in there and you need to bleed it through the Mayevsky valve. Look under the sink at the visible pipes - any leaks, rust stains, white buildup. If the radiators are cold, leaking, or you hear weird sounds in the pipes when the heating kicks on - call a technician. You can only bleed the air yourself; everything else is better left to a pro.